A closed organ pipe of length `L ` is in resonance with a tuning fork. If a hole is made in the pipe at a distance `L/4` from closed - end, it will be in resonance again, when:

Options

A. Tuning fork is replaced by another of high frequency
B. Tuning fork is replaced by another of lower frequency.
C. It will be resonance with same tuning fork
D. Now the pipe will never resonate with any tuning fork.

Answer

To solve the problem, we need to analyze the situation of a closed organ pipe and the effect of making a hole in it. Here's a step-by-step solution:

### Step 1: Understand the Closed Organ Pipe
A closed organ pipe is closed at one end and open at the other. The fundamental frequency (first harmonic) of a closed pipe is given by the formula:
\[ f_1 = \frac{v}{4L} \]
where \( v \) is the speed of sound in air, and \( L \) is the length of the pipe.

### Step 2: Identify the Effect of Making a Hole
When a hole is made at a distance \( L/4 \) from the closed end, the pipe can be thought of as being divided into two sections:
1. The section from the closed end to the hole (length = \( L/4 \)) - this behaves like a closed pipe.
2. The section from the hole to the open end (length = \( 3L/4 \)) - this behaves like an open pipe.

### Step 3: Analyze the New Pipe Sections
- The first section (closed pipe of length \( L/4 \)) will have a fundamental frequency given by:
\[ f_{closed} = \frac{v}{4 \times (L/4)} = \frac{v}{L} \]

- The second section (open pipe of length \( 3L/4 \)) will have a fundamental frequency given by:
\[ f_{open} = \frac{v}{2 \times (3L/4)} = \frac{v}{(3/2)L} = \frac{2v}{3L} \]

### Step 4: Determine Resonance Conditions
For the entire pipe to resonate, the frequencies of both sections must be the same. However, we can see that:
- \( f_{closed} = \frac{v}{L} \)
- \( f_{open} = \frac{2v}{3L} \)

Since these two frequencies are not equal, resonance cannot occur in this configuration.

### Step 5: Conclusion
Since the two sections of the pipe do not resonate together due to their differing frequencies, the pipe will not resonate with any tuning fork after the hole is made.

### Final Answer
The correct option is that the pipe will never resonate with any tuning fork.

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